where \(\mu \in \mathbb {R}\)  and \(2<q<2^*=\frac{2N}{(N-2)^+}\) . If \(\mu >0\) , it was proved in [1]} that if the product of \(\mu \)  and the mass is well controlled, there is a normalized ground state to (REF ). In addition, Soave [1]} raised an open question on how to obtain the second normalized solution to (REF ) since the associated energy functional constrained on the \(L^2\) -sphere admits a convex-concave geometry. It is worth pointing out that, Jeanjean, Le [3]} solved this open question for \(N \ge 4\) . Recently, Wei and Wu [4]} proved the existence of normalized solutions of mountain-pass type of (REF ) for \(N\ge 3\)  and \(2<q<2+\frac{4}{N}\) . Moreover, in [4]}, they also got the precisely asymptotic behaviors of ground states and mountain-pass solutions as \(\mu \rightarrow 0\)  or \(\mu \)  goes to its upper bound. These studies answer some open questions proposed by Soave in [1]}.
